# Number of Exponents

Let us take a simple number â€“ 10

^{5}^{}This is read as â€“ 10 to the power 5, or we say that exponent of 10 is 5.

In simple terms, exponents are also known as Power.

Example-1

What is the maximum value of

*s*if*N*= (35 Ã— 45 Ã— 55 Ã— 60 Ã— 124 Ã— 75) is divisible by 5*?*^{s}Solution

If we factorize

*N*= (35 Ã— 45 Ã— 55 Ã— 60 Ã— 124 Ã— 75), then we can see that 5 appears 6 times, it means*N*is divisible by 5^{6}.So, maximum value of

*s*= 6Exponent of any prime number P in

*n*! = , where*n*â‰¥ p*and [.] denotes the greatest integer value i.e., we have to consider only the integral value.*^{x}Let us find the exponent of 5 in 1000! = 1000/5 + 1000/5

^{2}+ 1000/5^{3 }+ 1000/5^{4}= 200 + 40 + 8 + 1 = 249Example-2

What is the highest power of 5 which can divide

*N*= (22! + 17894!)?Solution

Number of times this number is divisible by 5 is same as the number of zeroes at the end of this number. Since 22! have 4 zeroes at its end, so

*N*will also be having only four zeroes at its end. Hence, highest power of 5, which can divide*N*is 4.

# Process to find out the exponent of any composite number in n!

There are three different kinds of composite numbers.- Product of two or more than two prime numbers with unit power of all the prime numbers
- (Any prime number)
, where^{n}*n*>1^{2}), 27 (3^{3}) - Product of two or more than two prime numbers with power of any one prime number more than 1.
^{2 }Ã— 3), 72 (2^{3 }Ã— 3^{2}) etc.

Let us find out the exponents of the above-written composite numbers one by one:

- Find out the exponent of 15 in 100!

15 is the product of two distinct prime numbers 5 and 3.

So, to find out the exponents of 15, we need to find out the exponents of 5 and 3 individually.
So, by applying the same formula of finding out the exponents for any prime number in both of these cases individually, minimum of those two will be the answer.

100/5

*= [100/5] + [100/5*^{x}^{2}] = 20 + 4 = 24100/3

*= [100/3] + [100/3*^{x}^{2}] + [100/3^{3}] + [100/3^{4}]= 33 + 11 + 3 + 1 = 48

Obviously, 24 is going to be the answer.

- Find out the exponent of 25 in 100

25 = 5^{2}

^{In this case, we first find out the exponents of 5 and then divide it by 2 (actually the power) to find the exponents of 25.}

^{100/5x = [100/5] + [100/52] = 20 + 4 = 24}

^{So, 100/25x = 24/2 = 12} - Similarly, we can find out for the third category of numbers also.